The Split Delivery Capacitated

Team Orienteering Problem

C. Archetti, N. Bianchessi, A. Hertz,

M.G. Speranza

ROUTE 2011, Sitges (Spain), May 31-June 3, 2011

Motivation

In Europe a large percentage of trucks travel with no load (30%-40%)

Average load around 1/3 of the capacity

Small size of the carriers

Geographic dispersion of the customers

Low level of cooperation among carriers

Customers service level

Opportunities

Increase of the size of a carrier

Acquisition of other carriers

Major issue

Which carriers to acquire?

Which carriers ‘complement’ the fleet at best?

Opportunities

Cooperation among carriers

Exchange of customers (a customer may be inconvenient for a carrier but convenient for another one)

Major issue

How to cooperate?

Which customers to exchange?

Opportunities

Electronic interaction between shippers and carriers

Electronic database of spot demands by shippers

Electronic auctions

Major issue

How to choose the customers from a database?

On which customers to bid in an auction?

Opportunities

Dynamic pricing

Attract customers from low demand areas

Major issue

How to dynamically price the service?

How profitable are additional customers?

Given a set of potential customers, which are the most profitable ones?

How to price new customers?

General issues

Routing problems with profits?

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X

XX

X

Routing problems with profits

New customers – additional profit

TSP with profits

Orienteering problem (OP)

Max collected profit (constraint on time)

One uncapacitated vehicle - Feillet, Dejax, Gendreau (2005)

Team Orienteering Problem (TOP)

Potential customers – associated profitFleet of vehicles - maximum time available for each tour

Objective: Maximize the total profit

Capacitated TOP (CTOP) – vehicles are capacitated

The literature

Definition Butt, Cavalier (1994) Chao, Golden, Wasil (1996)

Heuristic Algorithms Tsiligirides (1984): Heuristic algorithm for the OP Chao, Golden, Wasil (1996): Heuristic algorithm for the TOP Tang, Miller-Hooks (2005): Tabu search + Adaptive memory Archetti, Hertz, Speranza: (2007) Tabu search and VNS Ke, Archetti, Feng (2008): Ant colony

Exact Algorithms Boussier, Feillet, Gendreau (2007): Branch-and-price

TOP

• Archetti et al (2009): Branch-and-price and heuristic• Archetti, Bianchessi, Speranza (2010): Branch-and-price and heuristic CTOP

Motivation for Split Deliveries

Possible increase of profit in CTOP?

The state of the art

SDVRP and variants

M. Dror, P. Trudeau, Transportation Sci., 1989 and Naval Research Logistics, 1990

M. Dror, G. Laporte, P. Trudeau, Discrete Applied Mathematics, 1994P.W. Frizzell, J.W. Giffin, Asia Pacific J. of Operations Research, 1992 and

Computers and Operations Research, 1995P.A. Mullaseril, M. Dror, J. Leung, J. Operational Research Society, 1997G. Sierksma, G.A. Tijssen, Annals of Operations Research, 1998C. Gueguen, PhD thesis, 1999J.M. Belenguer, M.C. Martinez, E. Mota, Operations Research, 2000

Before 2000

The state of the art

SDVRPC. Archetti, R. Mansini, M.G. Speranza, Transportation Sci., 2005K. Liu, PhD thesis, 2005C.G. Lee et al, Transportation Res. B, 2006C. Archetti, M. Savelsbergh, M.G. Speranza, Transportation Sci., 2006C. Archetti, A. Hertz, M.G. Speranza, Transportation Sci., 2006M. Jin, K. Liu, R.O. Bowden, IJPE, 2007M. Boudia, C. Prins, M. Reghioui, Lecture Notes in Computer Science, 2007S. Chen, B. Golden, E. Wasil, Networks, 2008C. Archetti, M. Savelsbergh, M.G. Speranza, Transportation Res. E, 2008C. Archetti, M. Savelsbergh, M.G. Speranza, Transportation Sci., 2008M. Jin, K. Liu, B. Eksioglu, Operations Research Letters, 2008L. Moreno, PhD thesis, 2008L. Moreno, M. Poggi de Aragao, E. Uchoa, Operations Research Letters, 2010U. Derigs, B. Li, U. Vogel, JORS, 2010 E. Mota, V. Campos, A. Corberan, working paper

R. E. Aleman, X. Zhang, R. R. Hill, J. Heuristics, 2010

R. E. Aleman, R. R. Hill, Int. J. Metaheuristics, 2010

After 2000

Time windowsS.C. Ho, D. Haugland, Computers & OR, 2004 D. Feillet et al, working paperG. Desaulniers, Operations Research, 2010

Pick-up and deliveryM. Nowak, PhD Thesis, 2005S. Mitra, APJOR, 2005M. Nowak, O. Ergun, C.C. White, Transportation Sci., 2008S. Mitra, JORS, 2008

Profit maximizationJ.E. Korsvik, K. Fagerholt, G. Laporte, working paperG. Bronmo, M. Christiansen, B. Nygreen, JORS, 2007

The state of the art - Variants

Inventory and productionY. Yu, H. Chen, F. Chu, Int. J. of Services Op. and Informatics, 2006 Y. Yu, H. Chen, F. Chu, EJOR, 2008M.C. Bolduc et al, EJOR, 2010

Minimum fraction servedB. Golden, D. Gulczynski, E. Wasil, Transportation Res. B, 2010

Heterogenous fleetR. Tavakkoli-Moghaddam et al, J. Franklyn Institute, 2007

Arc routing

N. Labadi, C. Prins, M. Reghioui, volume ‘Recent advances…’, 2008

J.M. Belenguer et, Transportation Sci., 2010

The state of the art - Variants

Real time

S.R. Thangiah, A.Fergany, S. Awan, CEJOR, 2007

Applications

S. Song, K. Lee, G. Kim, Comp. & Ind. Engineering, 2002

D. Ambrosino, A. Sciomachen, IMA J. of Man. Mathematics, 2007

P. Belfiore, H.T.Y. Yoshizaki, EJOR, 2010

The state of the art - Variants

The k-split cycles

Definition: Given any subset of k customers 1, 2, ..., k and k routes. Route 1 visits customers 1 and 2, route 2 visits customers 2 and 3, ..., route k−1 visits customers k−1 and k, and route k visits customers k and 1. The subset of customers 1, 2, ..., k is called a k-split cycle.

1

k

43

2

a k-split cycle

M. Dror, P. Trudeau, Transportation Sci., 1989

Some properties

Properties: If the cost matrix satisfies the triangle inequality, then there exists an optimal solution to the SDVRP where:

there is no k-split cycle (for any k); no two routes have more than one customer

with a split delivery in common; the number of splits is less than the number of

routes.

CTOP and SDCTOP

Is it worthwhile to allow split deliveries?

How much can be gained by split deliveries?

Qdi

Max. gain with split deliveries

2

1

SDCTOP

CTOP and the bound is tight

Tightness of the bound

12

,12

QQ

m vehicles, capacity

2m customers

Optimal solution CTOP:a direct trip to each customer.As m vehicles are available,

profit=mp

Optimal solution SDCTOP:m-1 vehicles visit 2 customers each

and 1 vehicle delivers the 2 missing unitsto m-1 customers.

profit=2(m-1)p

12

Q

mQ 2

demand =

profit = p

Algorithms

•Branch-and-price

•Matheuristic

Branch-and-price

Column generation identifies many good columns/routes

Use CPLEX to solve MILP exactly on subsets of routes

Identify good subsets of routes

Adapted from Archetti, Bianchessi, Speranza (2010) for SDVRP

Improved lower bound

A matheuristic

Heuristic or metaheuristic

scheme

Mathematical

programming

models

Matheuristic

A matheuristic for CTOP

Initialize generates initial solution

A Tabu search is run

Whenever a new best solution is found Optimize is run

Goal: to modify the current solution by inserting new customers or removing currently served customers to increase as much as possible the profit

Optimize

Binary variables:

insertion in a current route of a customerremoval of a customer from a current route

Continuous variables:

quantity delivered in each route to each served customer

Optimize

Delicate issues:

Complexity of the resulting modelGood estimation of the value of the improvement

Optimal solution of a MILP model

At most one customer can be removed

Tested instances

Known sets of benchmark instances, from 10 Christofides, Mingozzi, Toth (1979)

instances taken from the VRP library with both capacity and time constraintsNumber of vertices: from 51 to 200.

Set 1 - 10 instances - original instances

large number of vehicles

Set 2 - 90 instances

a smaller number of vehicles (m=2,3,4) and various values of Q and Tmax

Set 3 - 30 instances

changing the number of vehicles (m=2,3,4) with respect to the original values

A new set of instances

110 instances - 10 scenarios for each original instance

not interestingas all customers are served

Set 2

# of optimal solutions

Average optimality gap (%)

Branch-and-price

50/90 0.37

Matheuristic 42/90 0.51

Set 2(due to split deliveries)

Set 3

# of optimal solutions

Average optimality gap (%)

Branch-and-price

2/305.61

(over 20 instances)

Matheuristic 20/300.18

(over 20 instances)

Almost no improvement with split deliveries

Set 4

# of optimal solutions

Average optimality gap (%)

Branch-and-price

14/55 1.84

Matheuristic 27/55 0.22

Set 4

B&PMatheuristic

(due to split deliveries)

Conclusions

Routing problems with profits are an interesting class to explore

The branch-and-price can solve instances of reasonable size and provides optimality gaps

The heuristic use of the columns and matheuristic are both excellent directions to find high quality heuristic solutions